Continuous Improvement: Insights from 'Topology'

4TheMajor

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Very nice! Two small notes:

- The two notions of continuity (sequential continuity and topological continuity) you present under "Multivariate continuity" are
*not*equivalent. In a sense the topology around a point can be 'too large' to recover it from just convergence of sequences (in particular, these notions are equivalent for first countable spaces (I think? Second countability is definitely enough, but I think first countability also is) but not for general topological spaces). You can fix this by replacing the sequences with nets. - The compactifications (one-point and Stone-Cech) are very useful for classification and representation theorems, but personally I've hardly ever used them outside of that context. These compactifications are very deep mathematical results but also a bit niche.

I remember back when I took my course on Introduction to Topology that we spent a lot of time introducing homotopies and equivalence classes, and later the fundamental group. And then all that hard work paid off in a matter of minutes when Brouwer's fixed point theorem (on the 2-dimensional disc) was proven with these fundamental groups, which is actually one of the shorter proofs of this theorem if you already have the topological tools available.

Wrt continuity, I was implicitly just thinking of metric spaces (which are all first-countable, obviously). I’ll edit the post to clarify.

## Foreword

Sometimes you really like someone, but you can't for the life of you understand why. By all means, you should have tired of them long ago, but you keep coming back for more. Welcome, my friend, to

Topology.This book is a good one, but boy was it

slow(349 pages at ~30 minutes a page, on average). I just kept coming back, and I was slowly rewarded each time I did.Note: sil ver already reviewed

Topology.## Topology

Topology is about what it means for things to be "close" in a very abstract and general sense. Rather than taking on the monstrous task of intuitively explaining topology without math, I'm just going to talk about random things from the book and (literally) illustrate concepts which were at first confusing.

## Compactness = wonderful kind of mathematical "smallness"

Minimum description length says that an explanation is big if its shortest computational specification is long. You can have a simple explanation of a very long list of things or of a large universe, and extremely complicated explanations of things easily expressed in natural language (God's source code would be

a lotlonger than Maxwell's equations).VC dimension says a class of hypotheses is hard to learn if it has lots of predictive degrees of freedom. You can have an infinite class of hypotheses which is really easy to learn because it has low VC dimension (thresholding functions at value θ), and a finite class which is really hard to learn because it has high VC dimension (all C programs less than 1 million characters).

Compactness says that a topological space is big if it has a covering of open sets that can't be trimmed down to a finite subcollection which still covers the whole space. You can have an uncountable compact space ([0,1] under the standard topology, or even a Cantor space), and a countable space which isn't compact (Q under the standard topology; note that all countable topological spaces have to at least be Lindelof).

## Compactness is not always inherited by open subspaces

At first, I was confused why

opensubspaces Y of compact X don't have to be compact (if Y is closed, it does have to be compact). But compactness requiresallopen coverings of Y to have a finite subcover. Meaning, you can't just give it X's finite cover intersect the subspace, because the finite subcover has to be a subcollection of Y's covering.## Getting closure

Theorem: If X is compact, show that the projection π2:X×Y→Y is closed.I was confused why we needed compactness. Essentially, I didn't understand the tube lemma.

Now let's prove the theorem. Suppose C is closed in X×Y. We want to show f(C) is also closed. Take y∉π(C). (X×Y)−C is an open set of the domain containing the slice X×{y}. Since X is compact, apply the tube lemma to get a tube X×U. The projection of this tube is both open (because U is open in Y) and disjoint from π(C) (because the tube is contained in (X×Y)−C). Thus, all y∉π(C) have an open neighborhood disjoint from π(C), so π(C) must be closed.

Let X be a locally compact space. If f:X→Y is continuous, does it follow that f(X) is locally compact? What if f is both continuous and open?It has to be both continuous and open; the reason I got confused here was it seemed like continuity should be enough. It

wasplain to me how to prove it given f open, but this SE post has a good counterexample for just f continuous.## Multivariate continuity

How come you can have discontinuous multivariate functions which are continuous in each variable? What

iscontinuity, with a product space as your domain? To simplify matters, let’s consider two metric spaces X,Y.One definition of continuity uses open sets – f:X→Y is continuous at x if, for every open neighborhood U of f(x), there exists an open neighborhood V of x such that f(V)⊆U.

Another definition uses topological convergence. f:X→Y is continuous at x if, for every sequence xn→x, f(xn)→f(x).

These definitions are equivalent. The latter lets us think about how different winding paths you can take in a domain always must topologically converge to the same thing in the co-domain.

Continuity in the variables says that paths along the axes converge in the right way. But for continuity overall, we need

allpaths to converge in the right way. Directional continuity when the domain is R is a special case of this: continuity from below and from above if and only if continuity for all sequences converging topologically to x.## You only lift once

Suppose p:C→Y is a covering map. One way of understanding lifts in algebraic topology is that, for some path f:X→Y, the lift ~f:X→C is the unique path in the covering space C corresponding to f=p∘~f.

Once you fix the initial point, the lift corresponds to the unique path in the covering space which produces f. It's just helping you find the corresponding path in the lifted up covering space!

## Homotopy

This concept yields amazing insight into such profound topics as the deeper nature of jump rope. Under the standard subspace topology of R3, consider the space swept out by a rope held at fixed endpoints and tautness. All paths between the endpoints are path homotopic! You can think about movements of the rope (either clockwise or counterclockwise) as homotopies in this space.

## Miscellaneous

I stopped at about section 56 because I was getting diminishing returns. By this point, I felt like I had a solid understanding of point-set topology, and look forward to more thoroughly covering algebraic topology in the future.

One-point compactifications feel like an important thing to grasp, and they're fun to play around with mentally. I skipped Stone-Cech compactification.

Completeness in metric spaces means that Cauchy sequences converge topologically; in other words, nothing can "escape" from the space. I remember having problems with this (and with thinking about non-Hausdorff spaces) back when I was learning analysis. Things feel a lot better now.

## Verdict

Topologycan be dry, but it's exceedingly well-written and clear. I tried for quite a while to find a better topology book, but I didn't.## Forwards

Finally getting around to topology was such a good decision. For exercise solutions, see both MathOverflow and this site.

Some things change how you look at math, help you notice subtleties and shades and immediately grasp certain facets of new mathematical objects. Topology is one of these things, as is abstract algebra. Learning that an object is a group, or finitely generated, or isomorphic to a more familiar structure gives me an immediate head start. Similarly, learning that spaces are homeomorphic, or compact, or second-countable is

sucha boost.What was I even

doingwith my life before I knew about homeomorphisms?